XVI Avogadro Meeting - December 21st, 2020

String Theory Compactification

Superstrings defined in \(D = 10 \quad \Rightarrow \quad \mathcal{M}^{1, 9} = \mathcal{M}^{1, 3} \otimes X_6\)

Requirements

  • \(X_6\) is a compact manifold \((M, g)\)
  • \(N = 1\) SUSY in 4D
  • SM \(\subset\) arising gauge algebra

Solution

  • \(\dim_{\mathbb{C}} M = m\)
  • \(\mathrm{Hol}(g) \subseteq \mathrm{SU}(m)\)
  • \(\mathrm{Ric}(g) \equiv 0\) or \(c_1(M) \equiv 0\)

Calabi-Yau Manifolds

  • no known metric for compact CY
  • need to study topology (Hodge numbers) to infer 4D properties

\[h^{r,s} = \dim_{\mathbb{C}} \mathrm{H}_{\overline{\partial}}^{r,s}(M, \mathbb{C})\]

Complete Intersection Calabi-Yau Manifolds

Systems of \(k\) homogeneous equations from products of \(m\) projective spaces

\[ \sum\limits_{r = 1}^m p_{\alpha}^{i_r}\, \left( z_{i_r} \right)^{a^r_{\alpha}} = 0 \quad \rightarrow \quad X = \begin{bmatrix} \mathbb{P}^{n_1} & | & a^1_1 & \cdots & a^1_k \\ \vdots & & \vdots & \ddots & \vdots \\ \mathbb{P}^{n_m} & | & a^m_1 & \cdots & a^m_k \end{bmatrix} \]

such that \[ \begin{matrix} \text{degree of eq.} & \text{dim. of CY} & c_1 \equiv 0 \\ \Downarrow & \Downarrow & \Downarrow \\ a^r_{\alpha} \in \mathbb{N} & \dim_{\mathbb{C}} X = \sum\limits_{r = 1}^m n_r - k = 3 & n_r + 1 = \sum\limits_{\alpha = 1}^k a^r_{\alpha} \end{matrix} \] where \(a^r_{\alpha}\) are powers of coordinates on \(\mathbb{P}^{n_r}\) in equation \(\alpha\).

Available Data

\(\exists\) compiled datasets of 7890 CICY 3-folds with all Hodge numbers [Green et al. (1987)]

Supervision and Function Approximation

configuration matrix \(\quad \longrightarrow \quad\) \(\mathcal{R}\colon \quad \mathbb{N}^{m \times k} \longrightarrow \mathbb{N}\) \(\quad \longrightarrow \quad\) Hodge numbers

SUPERVISED LEARNING

  • replace \(\mathcal{R}( X )\) with \(\mathcal{R}_n( X;\, W )\)

    • \(W\) are controlled and tuned
    • optimisation problem

  • feed the algorithms \(X\) and \(h^{p,q}\) (true values)

  • get \(W\) such that \(\exists n > M > 0\) such that \[ \mathcal{L}_n(h^{p,q},\, \widehat{h^{p,q}}) < \epsilon, \quad \forall \epsilon > 0 \]

  • follow gradient descent of \(\mathcal{L}\) \(\rightarrow\) tune \(W\)

Neural Networks as Function Approximators

FULLY CONNECTED NETWORKS

  • older in design [Rosenblatt (1958)]
  • analogy neuron - “perceptron”
  • rose to fame in the 70s/80s
  • simple matrix multiplications \(+\) non linearity \[ a^{\{l+1\}} = \phi\left( w^{\{l\}} a^{\{l\}} + b^{\{l\}} \mathbb{I} \right) \in \mathbb{R}^p \] \[ \phi(z) = \mathrm{ReLU}(z) = \max(0, z) \]
  • backpropagation \(\rightarrow\) \(W^{\{l\}} = \left( w^{\{l\}}, b^{\{l\}} \right)\)

CONVOLUTIONAL NEURAL NETWORKS

  • newer in conception [LeCun et al. (1989)]
  • based on sliding windows (aka convolutions) \[ a^{\{l+1\}} = \phi\left( w^{\{l\}} * a^{\{l\}} + b^{\{l\}} \mathbb{I} \right) \in \mathbb{R}^{p \times q} \]
  • less parameters to isolate features

Inception Neural Networks

  • inspired by Google [Szegedy et al. (2014)]
  • created for computer vision
  • \(2 \times 10^5\) parameters (vs \(\ge 2 \times 10^6\))
  • concurrent kernels \(\Rightarrow\) shared parameters
  • retains “spatial awareness”
  • improved generalisation ability

Results

[Erbin, RF (2020)]

Comparison

[Erbin, RF (2020)]

Conclusion

[Machine learning is]
the field of study that gives computers the ability to learn without being explicitly programmed.

A. Samuel (1959)

  • deep learning can be a reliable predictive method
  • it can be used as source of inspiration for inference and generalisation
  • CNNs have a lot of unexpressed potential in physics (first time?)
  • the approach intersects mathematics, physics and computer science

What Lies Ahead?

  • improve \(h^{2,1}\) and “un-blackbox” the model (SHAP, filter analysis, etc.)
  • exploration for CICY 4-folds and representation learning (DGAN, VAE, etc.)
  • study symmetries (GNNs, Transformers, etc.) and explore the sting landscape

THANK YOU